Hydrogen evolution reaction following the Slater–Pauling curve: acceleration of rate processes induced from dipole interaction between protons and ferromagnetic catalysts

Developing new concepts to design noble-metal-free catalysts is necessary to achieve the hydrogen economy and reduce global CO2 emissions. Here, we provide novel insights into the design of catalysts with internal magnetic fields by investigating the relationship between the hydrogen evolution reaction (HER) and the Slater–Pauling rule. This rule states that adding an element to a metal reduces the alloy's saturation magnetization by an amount proportional to the number of valence electrons outside the d shell of the added element. We observed that rapid hydrogen evolution occurred when the magnetic moment of the catalyst was high, as predicted by the Slater–Pauling rule. Numerical simulation of the dipole interaction revealed a critical distance, rC, at which the proton trajectory changes from a Brownian random walk to a close-approach orbit towards the ferromagnetic catalyst. The calculated rC was proportional to the magnetic moment, consistent with the experimental data. Interestingly, rC was proportional to the number of protons contributing to the HER and accurately reflected the migration length for the proton dissociation and hydration and the O–H bond length in water. The magnetic dipole interaction between the nuclear spin of the proton and the electronic spin of the magnetic catalyst is verified for the first time. The findings of this study will open a new direction in catalyst design aided by an internal magnetic field.

Bennett et al. 25 reported that magnetic properties are one of the intrinsic differences between WC and Pt. WC is nonmagnetic, whereas Pt exhibits high magnetic susceptibility. In our previous study, 36 an HER catalyst was developed by doping tungsten carbide (WC) lattice with ferromagnetic Co nanocrystals. The resulting alloy was used for the catalytic hydrolysis of ammonium borane (NH 3 BH 3 ), a material known as a highcapacity hydrogen-storage compound. [37][38][39][40][41][42][43] The activity of the novel carbide was 30% higher than Pt nanoparticles under the same conditions. We hypothesised that the enhanced catalytic activity was attributed to the synergistic effect of the WC matrix promoting hydrolytic cleavage of NH 3 BH 3 and the ferromagnetic Co crystals interacting with the nucleus spin of the protons. In the present study, we aimed to verify the interaction between the nucleus proton and magnetic substances from both experimentation and numerical simulation. The relationship between the rate of HER and the magnetic moment of a catalyst was investigated considering the Slater-Pauling rule. 44,45 This rule states that adding an element to a metal reduces the alloy's saturation magnetization by an amount proportional to the number of valence electrons outside the d shell of the added element. To understand the relationship between HER and the Slater-Pauling rule, 44,45 the dipole interaction 44 between the proton nucleus spin and magnetic substances was simulated based on the electromagnetism for the rst time.
An application of HER catalysts is the generation of hydrogen fuel from NH 3 BH 3. In its stable crystal form, NH 3 BH 3 contains 19.6 wt% hydrogen, [37][38][39][40][41][42][43] and is being investigated for efficient transportation of hydrogen-based fuel for portable fuelcell systems. Previous studies investigated the HER by hydrolysis over 10 wt% Co (ref. 39) or 2 wt% Pt (ref. 40) (both supported by Al 2 O 3 ) and found that the HER in the latter was signicantly faster than that in the former. A similar HER in NH 3 BH 3 (aq) catalysed by Ni nanoparticles (NPs) supported by a zeolite molecular sieve was observed. 42 The atomic conguration in the AB molecule was investigated by neutron diffraction. 38 The chemical bonding states for AB were investigated by rst principles calculations 41 and so X-ray adsorption spectroscopy. 43 The standard enthalpy of formation, D f H m ; at 298.15 K was determined by combustion calorimetry. 37 The catalytic design of selecting metals and alloys based on thermodynamic cycles for hydrolysis of AB have not previously been investigated. In the present study, a new strategy was developed to enhance catalytic performance and investigate the effect of an internal magnetic eld on the thermodynamic mechanism of the hydrolysis of AB. For all samples (except 1, 7, and 13), the corresponding precursor solutions were mixed to form homogeneous aqueous solutions with a total volume of 200 ml. The homogeneous mixtures of metal complexes were prepared by calcination at about 500 K. These mixtures were then thermally decomposed under O 2 at 773 K for 2 h to obtain homogeneous oxidecontaining components. Aer thermal decomposition, powder samples were produced by reducing oxide-containing components with H 2 at 1073 K for 2 h followed by cooling with 20 K min −1 . Using an electron probe microanalysis system (JXA-8530FPlus, Co., JOEL Ltd, Tokyo) with a 15 kV accelerating voltage, the homogeneous chemical compositions of the samples were conrmed via X-ray images. Particle sizes and morphologies were compared by scanning electron microscopy (SEM).

Sample preparation
The SA of the samples was determined by the Brunauer-Emmett-Teller method using nitrogen physisorption isotherms at 77 K, obtained with a sorption and porosity analyzer (BEL-SORP mini, MicrotracBEL Corp). The SA of the sample powders was about 2 m 2 g −1 , as shown in Table 1.

HER analysis
For HER tests, 20 mg of each sample was placed in a glass test tube containing 1 ml of H 2 O. Separately, solutions of NH 3 -BH 3 (aq) were made by dissolving 0.5 mmol NH 3 BH 3 (cr) in 1.5 ml H 2 O. This solution was then mixed with each sample to initiate hydrolysis and hydrogen evolution. The volume of evolved hydrogen (V HER ) was measured as a function of time t, and the hydrogen evolution rate (R HER ) was determined from the slope of the V HER vs. t curve. From this, it was determined that V HER increased linearly while excess unreacted NH 3 BH 3 (aq) remained. The used aqueous solution was decanted aer the rst V HER measurement, while the ferromagnetic samples were held back inside the test tube with an external magnet placed outside the glass test tube. Aer draining, the second and third V HER measurements were conducted similarly to the rst. Aer completing the third measurement, fresh water was added to the ferromagnetic samples, which were then drained. Aer drying the samples in a dryer overnight, a fourth measurement was performed in the same way. The fourth V HER measurement was similar to the rst three. Because the produced hydrogen gas appeared to reduce the sample surface, the V HER obtained in all four was averaged and used as the total V HER . During the HER test, the AB solution was stirred with a rotating magnetic stirrer.

Results and discussion
Hydrogen evolution following the Slater-Pauling magnetic rule The structures of the samples determined from XRD are summarized in Table 1. The phases were consistent with the phase diagrams 46 of the Co-Fe and Co-Ni binary systems. However, the phases composed in 85 mol% Co-15 mol% Fe (no. 4), 96 mol% Co-4 mol% Fe (no. 6), 96 mol% Co-4 mol% Ni (no. 8), 96 mol% Co-8 mol% Ni (no. 9), 85 mol% Co-15 mol% Ni (no. 10) were different from the equilibrium phase diagrams 46 as shown in Fig. S1-S5 (ESI †). In no. 4, the high temperature fcc solid solution 47 was only formed due to rapid cooling (20 K min −1 ), which prevented the formation of the equilibrium bcc solid solution. 46 In no. 6, the high temperature fcc solid solution 47 remained due to rapid cooling, which prevented the formation of the equilibrium hcp solid solution. 48 In no. 8-10, the high fcc solid solution 47 remained partly mixed with the equilibrium hcp solid solution 48 resulting from rapid cooling. However, the meta stable fcc solid solutions were in the ferromagnetic phase. Fig. 1a and b compare the plots of R HER for the hydrolysis of ammonium borane (NH 3 BH 3 ) 36 as a function of the valence electron concentration E conc , of the catalyst and the Slater-Pauling curve. 44,45 The composition, E conc , magnetic moment, SSA, and crystal structure of Fe, Co, Ni, and their binary alloys are shown in Table 1. The perpendicular axis of the Slater-Pauling curve 44,45 is the magnetic moment M Ferro with the units of Bohr magneton. The M Ferro of pure Fe, Co, and Ni, as well as their alloys, varies with E conc . The Co-Fe alloy, with an E conc of 8.33, has the highest M Ferro . The R HER increases with E conc , beginning with Fe (E conc = 8), and reaching a maximum for the 92Co-8Fe (mol%) alloy with E conc = 8.92. As E conc continues to increase, R HER decreases. Specically, R HER varies according to the Slater-Pauling curve, and a magnetic-moment-induced increase in R HER is observed.
To clarify the reason for that the magnetic-moment increase in R HER , the rst of all, the thermodynamic cycle of hydrolysis of AB was discussed. Table 2 shows the thermodynamic cycle of the hydrolysis of AB where the standard enthalpies of formation, D f H m ; at 298.15 K of the standard substances of NH 3 -BH 3 (cr), 37 H 2 O(l), 49 orthoboric acid (B(OH) 3 (aq)), 49 ammonium (NH 4+ (aq)), 49 metaboric acid (BO 2 − (aq)), 50 and H 2 (g) 49 are summarised in Table S1. † Eqn (I) shows the hydration reaction of NH 3 BH 3 (cr), where the thermodynamic value is unknown. Eqn (II) shows the HER of the hydrolysis of NH 3 BH 3 (aq). Eqn (III), rewritten as the sum of eqn (I) and (II), indicates the HER from the initial substance of NH 3 BH 3 (cr). Eqn (IV) shows the formation of NH 4+ (aq). Eqn (V) shows the formation of BO 2 − (aq). Finally, eqn (VI), rewritten as the sum of eqn (III)-(V), shows the nal state of the hydrolysis Since eqn (IV)-(VI) are spontaneous reactions, the HER is given by eqn (III). As the standard entropy, S m ; of NH 3 BH 3 (cr) has not yet been measured, the standard entropy of reaction, D r S°, and the standard Gibbs energy of reaction, D r G°, are unknown. However, D r G°is more negative than D r H°as the HER increases D r S°. Therefore, when a driving energy is applied corresponding to the hydrogen overpotential of metals, the HER reaches equilibrium, as dened by eqn (VI) via eqn (III).
Next, we sought to understand the HER mechanism using the Slater-Pauling rule. 44,45 Fig. 2 depicts a schematic illustration of the catalytic HER over a magnetic metal single domain, which is relevant for Fe, Co, and Ni pure metals and their alloys. The HER is carried out through the following steps: (1) NH 3 BH 3 molecules collide during Brownian motion 51 6) One e − is released, along with a proton, during the decomposition of BH 3 , while the other e − is released from OH − (aq) during coordination to form B(OH) 3 (aq). In other words, the source of charge transfer is the B atom adsorbed on the metal single domain. It is likely that an attractive dipole interaction 44 occurs directly between a proton and ferromagnetic single domain when the nuclear and electronic spins of the catalyst are aligned in parallel. As a result, the magnetic force of a ferromagnetic material interacts with protons followed by H 2 (g) evolution. An attractive dipole interaction 44 was studied by a numerical simulation in the next section.
The thermodynamic cycle shown in Table 2 can be re-written as an electrochemical cycle, given by Cathode: 6H + (aq) + 6e − = 3H 2 (g) Therefore, the cathodic protection mechanism prevents the surface of the sample from corrosion, which is further evidence for HER following the Slater-Pauling magnetic rule. 44 The maximum M Ferro in the Slater-Pauling curve is at E conc = 8.33, whereas the maximum R HER is at E conc = 8.92. That is, the HER drop was observed in the samples including 15 mol% Fe (no. 4), 35 mol% Fe (no. 3), 50 mol% Fe (no. 2), and pristine Fe (no. 1). The cause of the HER drop was examined. The NH 3 -BH 3 (aq) solution was alkaline (pH 8.53). The electric potential (E h )-pH diagram 52 indicates that Fe 2 O 3 (cr) is stable. Therefore, the aqueous ion equilibrated with Fe 2 O 3 (cr) is ferric hydroxide ion Fe 3+ (aq), which appears with an orange colour. Fig. 3 shows the aqueous solution aer rst series of the hydrolysis of AB over the sample 85 mol%-15 mol% Fe (no. 4). Orange colour development was clearly observed. The same colour development occurred with other samples including 35 mol% Fe (no. 3), 50 mol% Fe (no. 2), and pristine Fe (no. 1). Consequently, the corrosion of samples with over 15 mol% Fe resulted in overriding of the cathodic protection (eqn (2)).  It is well known that d-electron vacancies 53 have control on the corrosion of transition metals and alloys. d-electron vacancies 53 capture the electrons of OH − (aq), resulting in adsorption of the radical oxygen atoms. The alloys with high Fe content have many electron vacancies in the 3d band. The samples with over 15 mol% Fe (no. 1-4) had many d-electron vacancies, which adsorb the radical oxygen atoms to form a carrion product, Fe 2 O 3 . This product overrides the cathodic protection (eqn (2)) during the hydrolysis of AB, which was concluded to be the reason for the HER drop.
The quantitative contents of the d-electron vacancies 53 were dened as the N v values. 54 The phase stabilities of the super heat resistant alloys are estimated in terms of the N v values. The alloys with excess N v values form harmful s phases, as was determined by multiple regression analysis 54 of the experimental data. The optimum alloy compositions are simulated to be less than the critical N v value, known as PHACOMP (Phase Computation). 54 The d-orbital level parameter, M d , was suggested based on the molecular orbital calculations to update the N v values considering the alloying effect. 55 The N v (ref. 54) and M d (ref. 55) values for the samples are shown in Table 1. 54,55 The critical value of N v was 1.85 and M d was 0.17 eV for 85 mol% Co-8 mol% Fe (no. 4), which was determined to prevent corrosion. When values less than the critical values are used, HER is actively caused by the cathodic protection of the surface from corrosion.

Numerical simulation of the dipole interaction between proton and magnetic catalyst
A hypothesis that an attractive dipole interaction occurs directly between a proton and ferromagnetic single domain when its nuclear and the electronic spin of the catalyst are aligned in parallel based on that HER follows the Slater-Pauling rule. In this section, the dipole interaction 44 was directly investigated by numerical simulation. Fig. 4 depicts a schematic of the most fundamental model. 44 The potential energy resulting from this magnetic dipole interaction U mag , is dened as follows.
where M Ferro and M P are the magnetic moments of the nano sphere single domain, [56][57][58][59] and the proton, respectively, m 0 is the permeability of free space, 4p × 10 −7 (H m −1 ). 44 The r datum is the distance between the N and S, or S and N poles of them. Accordingly, the magnetic force, F, exerted by the ferromagnetic single domain to the proton is dened as follows.
The acceleration, a, of the proton is determined by dividing F by the mass of the proton m P (1.67262171 × 10 −27 kg). 60 The present model includes a Co single domain sphere with an assumed diameter, d, of 60 nm to correlate it with the same domain in the WC lattice used in our previous study. 36 The magnetic moment per Co atom 44,45 is 1.7m B, and the number of moles, n, in this Co single domain is 1.6945 × 10 −17 based on the density of hcp Co (8.9 Mg m −3 ). 61 Therefore, the number of atoms, N, is 1.0205 × 10 7 . The magnetic moment of the Co single domain is N × 1.7m B per Co atom; 44,45 hence, M Ferro equals 2.0211 × 10 −22 Wb m. The M P of a proton 44 is 6.33 × 10 −33 Wb m.
These M Ferro and M P values were used in eqn (3) to simulate a as a function of r, and the resulting plot is shown in Fig. 5. When r is 2.33375 mm, a is 9.807 ms −2 , consistent with the gravitational acceleration, g. 60 Because g was obtained at the appropriate distance, the present simulation accurately estimated the magnetic dipole interaction.
To conrm that the proton is attracted to the magnetic force of the ferromagnetic single domain, its hydration was subsequently investigated. The hydration enthalpy DH + ad is −260.7 ± 2.5 (kcal mol −1 ), 62  the dissociation and hydration is 0.03-0.08 nm. 64 Moreover, the O-H bond length in the H 2 O(l) molecule is 0.097 nm. 64 Fig. 7 shows the effect of U mag of a Co single domain on a proton as a function of r (in the range of 0.03-0.06 nm) as described by eqn (3). With decreasing r, U mag shis to negative values. Remarkably, at 0.044725 nm, U mag becomes equal to −1.811 [aJ (H + ad ) −1 ], 62,63 which is the DH + ad . The r value at which U mag is equal to DH + ad is the critical distance r c , exactly. Therefore, as shown in Fig. 7, when r < r c (0.044725 nm), U mag becomes deeper than DH + ad , causing a proton to be attracted to the Co single domain. Specically, protons assemble on the Co single domain, and hydrogen gas rapidly evolves. (A) Initially, a proton accepts an electron to form a hydrogen atom, followed by adsorption (Volmer mechanism 2,3 ). (B) Subsequently, another proton accepts another electron and aggregates with a hydrogen atom to form a dimer molecule, and a hydrogen gas molecule H 2 (g) desorbs (Heyrovsky mechanism 2,4 ). (C) Selectively, two hydrogen atoms aggregate to form a dimer molecule, and a hydrogen gas molecule H 2 (g) dissociates (Tafel mechanism 5,6 ).
In contrast, when r is greater than r c , U mag becomes shallower than DH + ad , causing a proton to move towards an H 2 O molecule and form H 3 O + (aq) in Brownian motion. In conclusion, r c is the exact distance at which the trajectory of a proton changes from a random walk caused by Brownian motion to an approach orbit towards the Co single domain. Moreover, the calculated r c value (0.044725 nm) is within the range of l m of proton dehydration and rehydration (0.03-0.08 nm 64,66 ) and the O-H bond length (0.097 nm (ref. 62) in water. The effective particle number, N eff , of the proton, of which diameter, l P , is 8.751 × 10 −7 nm, 67 contributing to the frequency factor, A, in the Arrhenius equation 68 is likely to be proportional to r c .
The r c values for other single domains with the same compositions as the samples (Table 1) were calculated in the same way. Additionally, the M Ferro values were estimated by interpolating the Slater-Pauling curve. Fig. 8 depicts the calculated r c as a function of E conc , which shows that E conc is optimal when r c is 8.5. Theoretically, the contribution of N eff to   R HER is the highest at E conc = 8.5. However, as shown in Fig. 1, the maximum R HER is obtained when E conc is 8.92. Due to excess Fe, the formation of Fe 2 O 3 , resulting from excess 3d electron vacancies as above described, inhibited by the Heyrovsky 2,4 and Tafel mechanisms, 5,6 which are mediated by the Volmer mechanism. 2,3 The auxiliary data for the present numerical simulation were summarized in Table 3.
The singular catalytic activity of the Co nanocrystal doped WC in our previous study 36 was discussed. The valence electron number of WC and Pt are the same. However, their magnetic properties are different and Pt has a higher magnetic susceptibility because WC is non-magnetic. Singular WC does not show catalytic activity. In our previous study, 36 WC was doped with 60 nm diameter ferro-magnetic Co crystals to introduce an ordered-spin conguration, which showed a R HER value even higher than that of the Pt nanoparticles during the hydrolysis of AB. A hypothesis for the enhanced catalytic activity was attributed to the synergistic effect of the WC matrix promoting hydrolytic cleavage of NH 3 BH 3 and the ferromagnetic Co crystals interacting with the nucleus spin of the protons. The present veried attractive dipole interaction between protons and Co is evidence for the hypothesis of singular catalytic activity of WC arising from an internal magnetic eld.
It was previously hypothesised that a synergetic effect of WC breaking NH 3 BH 3 (aq) to form protons and the antiparallel alignment of the nuclear spins of protons and electronic spins in a single domain to increase the magnetic entropy. 36,69,70 Therefore, the 1s electronic spin of the hydrogen atoms absorbed by Pd (ref. 71) and adsorbed on Gd (ref. 72) induce disorder of their electron spin polarisation, thereby increasing the entropy of the system. By the above-described mechanism for HER, it is concluded the dipole attractive interaction between proton and magnetic catalyst enhances HER. Aer donating electrons by the Volmer mechanism, 2,3 electronic spins of that hydrogen atoms are likely to be aligned against spins of Co. This problem should be further investigated by molecular orbital calculations.
AB is hopeful hydrogen fuel. [37][38][39][40][41][42][43] The HER by hydrolysis over the 10 wt% Co nano particle supported on the SiO 2 nano particle (Co/SiO 2 ) was investigated by Xu and Chandra. 39 In order to compare HER property of the Co particle obtained in the present study with one of Co/SiO 2 by Xu and Chandra, 39 their normalised hydrogen evolution rate per unit area (m −2 ), R N HER , was evaluated. Table 4 shows their R N HER data. The experimental conditions for Co/SiO 2 were 50 percent more conc. in the concentration of AB, 25 times larger in SSA (=52 m −2 g −1 ), and 83 percent smaller in mass (3.4 mg) than ones for the present Co particle. Considering difference in the experimental conditions, the intrinsic R N HER data appears to be similar. The atomic conguration in the AB molecule was investigated by the neutron diffraction by Klooster et al. 38 The chemical bonding states for AB were investigated by the rst principles calculation by Banu et al., 41 and so X-ray adsorption spectroscopy by Niibe et al. 43 The D f H m datum at 298.15 K was determined by the combustion calorimetry by Shaulov. 37 The effect of an internal magnetic eld cooperating the cathodic protection on the thermodynamic mechanism of the hydrolysis of AB has been suggested in the present study for the rst time.
In our previous study for the Co doped WC, 36 the activation energy determined by the Arrhenius plots was found to be well consistent with the electrochemical hydrogen over potential of Co. 73 This means hydrogen gas H 2 (g) is evolved over the Co crystals via Heyrovsky 2,4 and Tafel mechanisms. 5,6 The R N HER datum of the Co crystals in the WC matrix was compared with that of the Co particle in the present study. Where the ratio of SSA of the fcc Co 36 nanocrystals against the WC matrix were hypothetically assumed to be as same as the ratio of their volumes. 61,74 Their volumes were calculated from their densities ( Table 3). Table 4 shows their R N HER data. The R N HER of the Co Table 3 Auxiliary data of the permeability of free space (m 0 ), mass (m p ) and diameter (l P ) of the proton, magnetic moment of a Co atom (b), densities (r) of Co and WC, magnetic moment, (M P ) of a proton, hydration enthalpy (DH + ad ), gravitational acceleration (g), the adopted data of numbers of moles (n), and Co atoms (N), magnetic moment (M Ferro ), in the 60 nm sphere Co single domain for the present study Table 4 Normalised hydrogen evolution rate per unit area, R N HER , of the Co particle in the present study, compared with the Co nano-crystal in WC matrix, 36 and the 10 wt% Co nano particle supported on g-Al 2 O 3 nano particle. 39 R N HER /(H 2 mmol min −1 m −2 ) n(AB)/mmol V(H 2 O)/mL Remarks crystal was found to be 25 times higher than that of the Co particle in the present study. This means that WC, of which electrons density of states are similar to Pt, adsorbs much amount of AB followed by decomposing the B-N bonding, and makes much protons dissociating from AB. Much protons are aligned on the Co crystals by the attractive dipole interaction, and 25 times higher R N HER is concluded to be caused. The parameters used in the present simulation are an extremely small scale power functions as, e.g., M Ferro /(Wb m) = 2.0211 × 10 −22 , and M P /(Wb m) = 6.33 × 10 −33 (Table 3) 44 the obtained r c datum reects the structure parameters for H 3 O + (aq) and H 2 O(l) molecules, and a proton migration (Fig. 6), indicating that the present numerical simulation was done with highly accuracy. Although a very fundamental model based on the sphere single domain magnetic structure 64-66 was adopted, this is a novel nding and likely to give impact to general science due to that proton in aqueous solution are associated with various phenomena in nature. In the future, the dipole interactions in the complicated magnetic force lines from the magnetic substances composed of the multi domain should be investigated.

Conclusions
The current study claried the magnetic dipole interactions between the nuclear spin of a proton and the electronic spin of a magnetic catalyst for the rst time. The conclusions are as follows: (1) the HER rate varied in accordance with the Slater-Pauling rule, resulting in a rapid rate of hydrogen evolution for catalysts with a high magnetic moment. (2) Numerical simulation of the magnetic dipole interaction revealed a critical distance at which the proton, as H 3 O + (aq), changes its path from a Brownian random walk trajectory to a close-approach orbit towards the ferromagnetic catalyst, in the same order as the migration length of the dissociation and hydration of the proton and O-H bond length in water. Consequently, this study provides novel insights into noble-metal-free catalyst design from the viewpoint of the internal magnetic eld.

Conflicts of interest
There are no conicts to declare.

Author contributions
M. M. conceived the idea and wrote the paper; Y. O. and R. F. synthesized the materials and conducted the hydrogen evolution tests and SA measurements; H. Y. reviewed and validated the work shown in the paper; and H. Y. rendered helpful discussions.